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D₇×3₋¹⁺² #67378

order	378
isoclinism	isoclinic class of D₇×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₁

Aliases

D₇×3₋¹⁺²

Splittings

Fitting, Hall	C₇×3₋¹⁺² ⋊ C₂
Sylow	C₇ ⋊ C₂×3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃²×D₇
commutator, socle, cosocle	C₃×C₇	C₂×C₃²
Fitting	C₇×3₋¹⁺²	C₂
prime radical	C₃²×D₇	C₃
hypercenter	3₋¹⁺²	D₇
hypocenter	C₇	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3)(4,5)(6,7)(9,12,13)(11,15,16);(1,2,4,6,7,5,3)(8,13,16,10,12,11,14,9,15)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×AGL(1,ℤ/9), C₂×D₇×3₋¹⁺², #10588044⋊C₂, …
maximal normal subgroup of	D₇×AGL(1,ℤ/9), C₂×D₇×3₋¹⁺², #10588044⋊C₂, …
maximal quotient of	3₋¹⁺²×Dic₇, C₂×D₇×3₋¹⁺², D₇×C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 31);